Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
20,500 |
A monthly cell phone plan costs $30 per month, plus $10 cents per text message, plus $15 cents for each minute used over 25 hours, and an additional $5 for every gigabyte of data used over 15GB. In February, Emily sent 150 text messages, talked for 26 hours, and used 16GB of data. How much did she have to pay?
**A)** $40.00
**B)** $45.50
**C)** $50.00
**D)** $59.00
**E)** $70.00
|
59.00
| 65.625 |
20,501 |
A self-employed individual plans to distribute two types of products, A and B. According to a survey, when the investment amount is $x$ (where $x \geq 0$) in ten thousand yuan, the profits obtained from distributing products A and B are $f(x)$ and $g(x)$ in ten thousand yuan, respectively, where $f(x) = a(x - 1) + 2$ ($a > 0$); $g(x) = 6\ln(x + b)$, ($b > 0$). It is known that when the investment amount is zero, the profit is also zero.
(1) Determine the values of $a$ and $b$;
(2) If the self-employed individual is ready to invest 5 ten thousand yuan in these two products, please help him devise an investment plan to maximize his profit, and calculate the maximum value of his income. (Round to 0.1, reference data: $\ln 3 \approx 1.10$).
|
12.6
| 92.1875 |
20,502 |
How many four-digit numbers are divisible by 17?
|
530
| 2.34375 |
20,503 |
My school's Chess Club has 24 members. It needs to select 3 officers: president, secretary, and treasurer. Each person can hold at most one office. Two of the members, Alice and Bob, will only serve together as officers. In how many ways can the club choose its officers?
|
9372
| 64.0625 |
20,504 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If the area of $\triangle ABC$ is $S$, and $2S=(a+b)^{2}-c^{2}$, then $\tan C$ equals \_\_\_\_\_\_.
|
- \frac {4}{3}
| 53.125 |
20,505 |
Given the parametric equation of line $l$ as $$\begin{cases} \left.\begin{matrix}x= \frac {1}{2}t \\ y=1+ \frac { \sqrt {3}}{2}t\end{matrix}\right.\end{cases}$$ (where $t$ is the parameter), and the polar equation of curve $C$ as $\rho=2 \sqrt {2}\sin(\theta+ \frac {\pi}{4})$, line $l$ intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $P$.
(1) Find the standard equation of line $l$ and the Cartesian coordinate equation of curve $C$;
(2) Calculate the value of $\frac {1}{|PA|} + \frac {1}{|PB|}$.
|
\sqrt {5}
| 0 |
20,506 |
Given the parabola $C:y^{2}=8x$ with focus $F$ and directrix $l$. $P$ is a point on $l$ and the line $(PF)$ intersects the parabola $C$ at points $M$ and $N$. If $\overrightarrow{{PF}}=3\overrightarrow{{MF}}$, find the length of the segment $MN$.
|
\frac{32}{3}
| 11.71875 |
20,507 |
Among the following statements, the correct one is:
(1) The probability of event A or B happening is definitely greater than the probability of exactly one of A or B happening;
(2) The probability of events A and B happening simultaneously is definitely less than the probability of exactly one of A or B happening;
(3) Mutually exclusive events are definitely complementary events, and complementary events are also mutually exclusive events;
(4) Complementary events are definitely mutually exclusive events, but mutually exclusive events are not necessarily complementary events;
(5) If A and B are complementary events, then A+B cannot be a certain event.
|
(4)
| 1.5625 |
20,508 |
Find the radius of the circle with the equation $x^2 - 8x + y^2 - 10y + 34 = 0$.
|
\sqrt{7}
| 85.9375 |
20,509 |
If the coefficient of $x^6$ in the expansion of $(x^2-a)(x+\frac{1}{x})^{10}$ is 30, then find the value of $\int_{0}^{a} (3x^2+1) \, dx$.
|
10
| 75.78125 |
20,510 |
Determine how many solutions the following equation has:
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-2^2)(x-4^2)(x-6^2)\dotsm(x-24^2)} = 0
\]
|
47
| 63.28125 |
20,511 |
Solve for $y$: $$\log_4 \frac{2y+8}{3y-2} + \log_4 \frac{3y-2}{2y-5}=2$$
|
\frac{44}{15}
| 89.0625 |
20,512 |
What is the average of all the integer values of $N$ such that $\frac{N}{84}$ is strictly between $\frac{4}{9}$ and $\frac{2}{7}$?
|
31
| 8.59375 |
20,513 |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. The area of the triangle is S, and it is given that 2$\sqrt {3}$S - $\overrightarrow {AB}$•$\overrightarrow {AC}$ = 0, and c = 2.
(I) Find the measure of angle A.
(II) If a² + b² - c² = $\frac {6}{5}$ab, find the value of b.
|
\frac{3 + 4\sqrt{3}}{4}
| 2.34375 |
20,514 |
In isosceles $\vartriangle ABC, AB = AC, \angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$ , respectively, so that $AE = 10, AF = 15$ . The area of $\vartriangle AEF$ is $60$ , and the area of quadrilateral $BEFC$ is $102$ . Find $BC$ .
|
36
| 50 |
20,515 |
It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating.
|
40
| 69.53125 |
20,516 |
Calculate the sum of the first six terms of the geometric series $3 + \left(\frac{1}{3}\right) + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^3 + \left(\frac{1}{3}\right)^4 + \left(\frac{1}{3}\right)^5$. Express your answer as a simplified fraction.
|
\frac{364}{81}
| 54.6875 |
20,517 |
Given the function $f(x)=kx^{2}+2kx+1$ defined on the interval $[-3,2]$, the maximum value of the function is $4$. Determine the value of the real number $k$.
|
-3
| 25.78125 |
20,518 |
Given two numbers selected randomly from the interval $[0,1]$, find the probability that the sum of these two numbers is less than $\frac{8}{5}$.
|
\frac{23}{25}
| 30.46875 |
20,519 |
Given the function $f(x) = \sin 2x - \cos \left(2x+\dfrac{\pi}{6}\right)$.
(1) Find the value of $f\left(\dfrac{\pi}{6}\right)$.
(2) Find the minimum positive period and the interval of monotonic increase of the function $f(x)$.
(3) Find the maximum and minimum values of $f(x)$ on the interval $\left[0,\dfrac{7\pi}{12}\right]$.
|
-\dfrac{\sqrt{3}}{2}
| 50 |
20,520 |
Find the greatest prime that divides $$ 1^2 - 2^2 + 3^2 - 4^2 +...- 98^2 + 99^2. $$
|
11
| 74.21875 |
20,521 |
Given an ellipse E: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ ($$a > b > 0$$) passing through point Q ($$\frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}$$), the product of the slopes of the lines connecting the moving point P on the ellipse to the two endpoints of the minor axis is $$-\frac{1}{2}$$.
1. Find the equation of the ellipse E.
2. Let $$F_1$$ and $$F_2$$ be the left and right focal points of E, respectively. A line l passes through point $$F_1$$ and intersects E at points A and B. When $$\overrightarrow{F_2A} \cdot \overrightarrow{F_2B} = 2$$, find the area of triangle $$ABF_2$$.
|
\frac{4}{3}
| 40.625 |
20,522 |
1. Convert the parametric equations of the conic curve $C$:
$$
\begin{cases}
x=t^{2}+ \frac {1}{t^{2}}-2 \\
y=t- \frac {1}{t}
\end{cases}
$$
($t$ is the parameter) into a Cartesian coordinate equation.
2. If the polar equations of two curves are $\rho=1$ and $\rho=2\cos\left(\theta+ \frac {\pi}{3}\right)$ respectively, and they intersect at points $A$ and $B$, find the length of segment $AB$.
|
\sqrt {3}
| 0 |
20,523 |
In my office, there are two digital 24-hour clocks. One clock gains one minute every hour and the other loses two minutes every hour. Yesterday, I set both of them to the same time, but when I looked at them today, I saw that the time shown on one was 11:00 and the time on the other was 12:00. What time was it when I set the two clocks?
A) 23:00
B) 19:40
C) 15:40
D) 14:00
E) 11:20
|
15:40
| 14.0625 |
20,524 |
Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve *Daily Challenge* problems $D_1, D_2, D_3$ . However, he insists that carrot $C_i$ must be eaten only after solving *Daily Challenge* problem $D_i$ . In how many satisfactory orders can he complete all six actions?
*Proposed by Albert Wang (awang2004)*
|
90
| 53.125 |
20,525 |
Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers.
|
200
| 29.6875 |
20,526 |
Given an increasing sequence $\{a_n\}$ with $2017$ terms, and all terms are non-zero, $a_{2017}=1$. If two terms $a_i$, $a_j$ are arbitrarily chosen from $\{a_n\}$, when $i < j$, $a_j-a_i$ is still a term in the sequence $\{a_n\}$. Then, the sum of all terms in the sequence $S_{2017}=$ ______.
|
1009
| 23.4375 |
20,527 |
Given an equilateral triangle ABC, a student starts from point A and moves the chess piece using a dice-rolling method, where the direction of the movement is determined by the dice roll. Each time the dice is rolled, the chess piece is moved from one vertex of the triangle to another vertex. If the number rolled on the dice is greater than 3, the movement is counterclockwise; if the number rolled is not greater than 3, the movement is clockwise. Let Pn(A), Pn(B), Pn(C) denote the probabilities of the chess piece being at points A, B, C after n dice rolls, respectively. Calculate the probability of the chess piece being at point A after 7 dice rolls.
|
\frac{21}{64}
| 0 |
20,528 |
Let $ABCDEFGH$ be a cube with each edge of length $s$. A right square pyramid is placed on top of the cube such that its base aligns perfectly with the top face $EFGH$ of the cube, and its apex $P$ is directly above $E$ at a height $s$. Calculate $\sin \angle FAP$.
|
\frac{\sqrt{2}}{2}
| 42.96875 |
20,529 |
The on-time arrival rate of bus No. 101 in a certain city is 90%. Calculate the probability that the bus arrives on time exactly 4 times out of 5 rides for a person.
|
0.32805
| 69.53125 |
20,530 |
Let $AB = 10$ be a diameter of circle $P$ . Pick point $C$ on the circle such that $AC = 8$ . Let the circle with center $O$ be the incircle of $\vartriangle ABC$ . Extend line $AO$ to intersect circle $P$ again at $D$ . Find the length of $BD$ .
|
\sqrt{10}
| 17.1875 |
20,531 |
Let $S = \{1, 2,..., 8\}$ . How many ways are there to select two disjoint subsets of $S$ ?
|
6561
| 53.90625 |
20,532 |
Points $D$ and $E$ are chosen on side $BC$ of triangle $ABC$ such that $E$ is between $B$ and $D$ and $BE=1$ , $ED=DC=3$ . If $\angle BAD=\angle EAC=90^\circ$ , the area of $ABC$ can be expressed as $\tfrac{p\sqrt q}r$ , where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. Compute $p+q+r$ .
[asy]
import olympiad;
size(200);
defaultpen(linewidth(0.7)+fontsize(11pt));
pair D = origin, E = (3,0), C = (-3,0), B = (4,0);
path circ1 = arc(D,3,0,180), circ2 = arc(B/2,2,0,180);
pair A = intersectionpoint(circ1, circ2);
draw(E--A--C--B--A--D);
label(" $A$ ",A,N);
label(" $B$ ",B,SE);
label(" $C$ ",C,SW);
label(" $D$ ",D,S);
label(" $E$ ",E,S);
[/asy]
|
36
| 39.0625 |
20,533 |
Rotate a square with a side length of 1 around a line that contains one of its sides. The lateral surface area of the resulting solid is \_\_\_\_\_\_.
|
2\pi
| 95.3125 |
20,534 |
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, and identical letters correspond to identical digits). The word "GUATEMALA" was the result. How many different numbers could Egor have originally written, if his number was divisible by 25?
|
20160
| 7.03125 |
20,535 |
While waiting at the post office, Lena moved 40 feet closer to the counter over a period of 20 minutes. At this rate, how many minutes will it take her to move the remaining 100 meters to the counter?
|
164.042
| 17.96875 |
20,536 |
Given that the asymptotes of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and the axis of the parabola $x^{2} = 4y$ form a triangle with an area of $2$, calculate the eccentricity of the hyperbola.
|
\frac{\sqrt{5}}{2}
| 25.78125 |
20,537 |
There are two types of products, A and B, with profits of $p$ ten-thousand yuan and $q$ ten-thousand yuan, respectively. Their relationship with the invested capital $x$ ten-thousand yuan is: $p= \frac{1}{5}x$, $q= \frac{3}{5} \sqrt{x}$. Now, with an investment of 3 ten-thousand yuan in managing these two products, how much capital should be allocated to each product in order to maximize profit, and what is the maximum profit?
|
\frac{21}{20}
| 7.8125 |
20,538 |
Given the function $f(x) = \sin(2x + \frac{\pi}{3}) - \sqrt{3}\sin(2x - \frac{\pi}{6})$,
(1) Find the smallest positive period of the function $f(x)$ and its intervals of monotonic increase;
(2) When $x \in \left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$, find the maximum and minimum values of $f(x)$, and the corresponding values of $x$ at which these extreme values are attained.
|
-\sqrt{3}
| 10.15625 |
20,539 |
Queen High School has $1500$ students, and each student takes $6$ classes per day. Each teacher teaches $5$ classes, with each class having $25$ students and $1$ teacher. How many teachers are there at Queen High School?
|
72
| 65.625 |
20,540 |
Given a hyperbola $F$ with the equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, let $F$ be its left focus. Draw a line perpendicular to one asymptote of the hyperbola passing through $F$, and denote the foot of the perpendicular as $A$ and the intersection with the other asymptote as $B$. If $3\overrightarrow{FA} = \overrightarrow{FB}$, find the eccentricity of this hyperbola.
A) $2$
B) $3$
C) $\sqrt{2}$
D) $\sqrt{3}$
|
\sqrt{3}
| 2.34375 |
20,541 |
Any five points are taken inside or on a rectangle with dimensions 2 by 1. Let b be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than b. What is b?
|
\frac{\sqrt{5}}{2}
| 96.875 |
20,542 |
What is the value of $102^{4} - 4 \cdot 102^{3} + 6 \cdot 102^2 - 4 \cdot 102 + 1$?
|
100406401
| 0 |
20,543 |
Let $a$, $b$, and $c$ be the roots of the equation $x^3 - 2x - 5 = 0$. Find $\frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2}$.
|
10
| 70.3125 |
20,544 |
Given the function $f(x) = \cos x \cdot \sin\left(\frac{\pi}{6} - x\right)$,
(1) Find the interval where $f(x)$ is monotonically decreasing;
(2) In $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. If $f(C) = -\frac{1}{4}$, $a=2$, and the area of $\triangle ABC$ is $2\sqrt{3}$, find the length of side $c$.
|
2\sqrt{3}
| 38.28125 |
20,545 |
Given that $\sin x= \frac {3}{5}$, and $x\in( \frac {\pi}{2},\pi)$, find the values of $\cos 2x$ and $\tan (x+ \frac {\pi}{4})$.
|
\frac {1}{7}
| 60.15625 |
20,546 |
How many numbers are in the list $-50, -44, -38, \ldots, 68, 74$?
|
22
| 54.6875 |
20,547 |
Given an arithmetic sequence $\{a\_n\}$, where $a\_1=\tan 225^{\circ}$ and $a\_5=13a\_1$, let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{(-1)^na\_n\}$. Determine the value of $S\_{2015}$.
|
-3022
| 64.84375 |
20,548 |
Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7.
|
4410
| 59.375 |
20,549 |
Given a rectangular painting that measures $12$ inches high and $16$ inches wide, and is placed in a frame that forms a border three inches wide on all sides, find the area of the border, in square inches.
|
204
| 51.5625 |
20,550 |
Given the function f(x) = |lnx|, and real numbers m and n that satisfy 0 < m < n and f(m) = f(n). If the maximum value of f(x) in the interval [m^2, n] is 2, find the value of $\frac{n}{m}$.
|
e^2
| 35.9375 |
20,551 |
In triangle $DEF$, $DE=130$, $DF=110$, and $EF=140$. The angle bisector of angle $D$ intersects $\overline{EF}$ at point $T$, and the angle bisector of angle $E$ intersects $\overline{DF}$ at point $S$. Let $R$ and $U$ be the feet of the perpendiculars from $F$ to $\overline{ES}$ and $\overline{DT}$, respectively. Find $RU$.
|
60
| 7.03125 |
20,552 |
Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction.
|
\frac{1}{12}
| 91.40625 |
20,553 |
If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \dfrac{7}{6}\cdot \ldots\cdot \dfrac{c}{d} = 12$, determine the sum of $c$ and $d$.
|
71
| 89.84375 |
20,554 |
Let $g(x) = 3x^4 + 2x^3 - x^2 - 4x + s$. For what value of $s$ is $g(-1) = 0$?
|
-4
| 87.5 |
20,555 |
Given that vertex E of right triangle ABE, where AE=BE, is in the interior of unit square ABCD, let R be the region consisting of all points inside ABCD and outside triangle ABE whose distance from AD is between 1/4 and 1/2. Calculate the area of R.
|
\frac{1}{8}
| 15.625 |
20,556 |
Let $A=\{m-1,-3\}$, $B=\{2m-1,m-3\}$. If $A\cap B=\{-3\}$, then determine the value of the real number $m$.
|
-1
| 94.53125 |
20,557 |
Given that $x$ and $y$ are distinct nonzero real numbers such that $x - \tfrac{2}{x} = y - \tfrac{2}{y}$, determine the product $xy$.
|
-2
| 39.84375 |
20,558 |
Given vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ with pairwise angles of $60^\circ$, and $|\overrightarrow{a}|=|\overrightarrow{b}|=|\overrightarrow{c}|=1$, find $|\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}|$.
|
\sqrt{2}
| 89.84375 |
20,559 |
Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$.
|
45
| 97.65625 |
20,560 |
A factory assigns five newly recruited employees, including A and B, to three different workshops. Each workshop must be assigned at least one employee, and A and B must be assigned to the same workshop. The number of different ways to assign the employees is \_\_\_\_\_\_.
|
36
| 53.90625 |
20,561 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $4\cos C \cdot \sin^2 \frac{C}{2} + \cos 2C = 0$.
(1) If $\tan A = 2\tan B$, find the value of $\sin(A-B)$;
(2) If $3ab = 25 - c^2$, find the maximum area of $\triangle ABC$.
|
\frac{25\sqrt{3}}{16}
| 28.90625 |
20,562 |
In the expansion of $\left(\frac{1}{ \sqrt[3]{x}} +2x \sqrt {x}\right)^{7}$, the coefficient of $x^{5}$ is $\_\_\_\_\_\_$.
|
560
| 85.15625 |
20,563 |
A bag contains three balls labeled 1, 2, and 3. A ball is drawn from the bag, its number is recorded, and then it is returned to the bag. This process is repeated three times. If each ball has an equal chance of being drawn, calculate the probability of the number 2 being drawn three times given that the sum of the numbers drawn is 6.
|
\frac{1}{7}
| 81.25 |
20,564 |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given the equation acosB - bcosA + 2c = 0, find the value of $\frac{tanA}{tanB}$.
|
-\frac{1}{3}
| 46.09375 |
20,565 |
Given $\{a_{n}\}$ is a geometric sequence, $a_{2}a_{4}a_{5}=a_{3}a_{6}$, $a_{9}a_{10}=-8$, then $a_{7}=\_\_\_\_\_\_$.
|
-2
| 59.375 |
20,566 |
Determine the value of \( n \) such that \( 2^7 \cdot 3^4 \cdot n = 10! \).
|
350
| 15.625 |
20,567 |
Three workshops A, B, and C of a factory produced the same kind of product, with quantities of 120, 60, and 30, respectively. To determine if there are significant differences in product quality, a stratified sampling method was used to take a sample of size n for inspection, with 2 samples taken from workshop B.
(I) How many samples should be taken from workshops A and C, and what is the sample size n?
(II) Let the n samples be denoted by $A_1$, $A_2$, ..., $A_n$. Now, randomly select 2 samples from these.
(i) List all possible sampling outcomes.
(ii) Let M be the event "the 2 selected samples come from different workshops." Calculate the probability of event M occurring.
|
\frac{2}{3}
| 57.8125 |
20,568 |
(In this question, 12 points) During a shooting training session, the probabilities of a shooter hitting the 10, 9, 8, and 7 rings are 0.21, 0.23, 0.25, and 0.28, respectively. Calculate the probability that the shooter in a single shot:
(1) Hits either the 10 or 7 ring;
(2) Scores below 7 rings.
|
0.03
| 92.96875 |
20,569 |
Given the sequence $\{a\_n\}$, if $a\_1=0$ and $a\_i=k^2$ ($i \in \mathbb{N}^*, 2^k \leqslant i < 2^{k+1}, k=1,2,3,...$), find the smallest value of $i$ that satisfies $a\_i + a_{2i} \geq 100$.
|
128
| 22.65625 |
20,570 |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$.
|
40\pi
| 25 |
20,571 |
Mady has an infinite number of balls and boxes available to her. The empty boxes, each capable of holding sixteen balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. How many balls in total are in the boxes as a result of Mady's $2010$th step, considering the procedure implies hexadecimal (base 16) operations rather than quinary (base 5)?
|
30
| 10.15625 |
20,572 |
In a "clearance game," the rules stipulate that in round \( n \), a dice is to be rolled \( n \) times. If the sum of the points of these \( n \) rolls is greater than \( 2^{n} \), the player clears the round.
(1) What is the maximum number of rounds a player can clear in this game?
(2) What is the probability that the player clears the first three rounds consecutively?
(Note: The dice is a fair cube with faces numbered \( 1, 2, 3, 4, 5, 6 \), and the point on the top face after landing indicates the outcome of the roll.)
|
\frac{100}{243}
| 0.78125 |
20,573 |
A point $(x,y)$ is a distance of 15 units from the $x$-axis. It is a distance of 13 units from the point $(2,7)$. It is a distance $n$ from the origin. Given that $x>2$, what is $n$?
|
\sqrt{334 + 4\sqrt{105}}
| 28.125 |
20,574 |
Let \(x\) and \(y\) be positive real numbers. Find the minimum value of
\[
\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 1000 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 1000 \right).
\]
|
-500000
| 81.25 |
20,575 |
A sports lottery stipulates that 7 numbers are drawn from a total of 36 numbers, ranging from 01 to 36, for a single entry, which costs 2 yuan. A person wants to select the lucky number 18 first, then choose 3 consecutive numbers from 01 to 17, 2 consecutive numbers from 19 to 29, and 1 number from 30 to 36 to form an entry. If this person wants to purchase all possible entries that meet these requirements, how much money must they spend at least?
|
2100
| 6.25 |
20,576 |
If there are $1, $2, and $3 bills in the board game "Silly Bills" and let x be the number of $1 bills, then x+11, x-18, and x+11+(x-18) = 2x-7 are the respective number of $2 and $3 bills, determine the value of x when the total amount of money is $100.
|
22
| 83.59375 |
20,577 |
Given that the sequence starts with 2 and alternates by adding 2 between consecutive terms, find the 30th term of this arithmetic sequence.
|
60
| 86.71875 |
20,578 |
A team of four students goes to LMT, and each student brings a lunch. However, on the bus, the students’ lunches get mixed up, and during lunch time, each student chooses a random lunch to eat (no two students may eat the same lunch). What is the probability that each student chooses his or her own lunch correctly?
|
1/24
| 17.96875 |
20,579 |
The even throws belong to B. Bons. Therefore, Bons wins only when the total number of throws, including the last successful one, is even. The probability of rolling a six is $\frac{1}{6}$. The probability of the opposite event is $\frac{5}{6}$. Hence, the probability that the total number of throws will be even is
Method 2: Denote by $p$ the probability of the event "B. Bons wins". This event can happen in one of two ways:
1) Initially, J. Silver did not roll a 6, and B. Bons immediately rolled a 6. The probability of this is $\frac{5}{6} \cdot \frac{1}{6} = \frac{5}{36}$.
2) In the first round, both Silver and Bons did not roll a 6. After this, the game essentially starts anew, and B. Bons wins with probability $p$. The probability of this development is $\frac{5}{6} \cdot \frac{5}{6} \cdot p = \frac{25}{36} p$.
Thus, $p = \frac{5}{36} + \frac{25}{36} p$, leading to $p = \frac{5}{11}$.
|
\frac{5}{11}
| 97.65625 |
20,580 |
What is the largest possible value for the sum of five consecutive even numbers, if 10 and 12 are included amongst the five numbers?
|
70
| 48.4375 |
20,581 |
Given \( w \) and \( z \) are complex numbers such that \( |w+z|=2 \) and \( |w^2+z^2|=8 \), find the smallest possible value of \( |w^3+z^3| \).
|
20
| 55.46875 |
20,582 |
Each side of the square grid is 15 toothpicks long. Calculate the total number of toothpicks used to construct the square grid.
|
480
| 62.5 |
20,583 |
Given $\cos(\alpha + \frac{\pi}{4}) = \frac{\sqrt{2}}{4}$, find the value of $\sin(2\alpha)$.
|
\frac{3}{4}
| 90.625 |
20,584 |
Determine the share of the Japanese yen in the currency structure of the NWF funds as of 01.07.2021 using one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Japanese yen as of 01.07.2021:
\[ JPY_{22} = 1213.76 - 3.36 - 38.4 - 4.25 - 226.6 - 340.56 - 0.29 = 600.3 \text{ (billion rubles)} \]
b) Determine the share of Japanese yen in the currency structure of NWF funds as of 01.07.2021:
\[ \alpha_{07}^{JPY} = \frac{600.3}{1213.76} \approx 49.46\% \]
c) Calculate by how many percentage points and in what direction the share of Japanese yen in the currency structure of NWF funds has changed over the period considered in the table:
\[ \Delta \alpha^{JPY} = \alpha_{07}^{JPY} - \alpha_{06}^{JPY} = 49.46 - 72.98 = -23.52 \approx -23.5 \text{ (p.p.)} \]
Second method:
a) Determine the share of euros in the currency structure of NWF funds as of 01.07.2021:
\[ \alpha_{07}^{\text{EUR}} = \frac{38.4}{1213.76} \approx 3.16\% \]
b) Determine the share of Japanese yen in the currency structure of NWF funds as of 01.07.2021:
\[ \alpha_{07}^{JPY} = 100 - 0.28 - 3.16 - 0.35 - 18.67 - 28.06 - 0.02 = 49.46\% \]
c) Calculate by how many percentage points and in what direction the share of Japanese yen in the currency structure of NWF funds has changed over the period considered in the table:
\[ \Delta \alpha^{JPY} = \alpha_{07}^{JPY} - \alpha_{06}^{JPY} = 49.46 - 72.98 = -23.52 \approx -23.5 \text{ (p.p.)} \]
|
-23.5
| 67.1875 |
20,585 |
Given that out of 8 teams, there are 3 weak teams, these 8 teams are divided into two groups $A$ and $B$ with 4 teams in each group by drawing lots.
1. The probability that one of the groups $A$ or $B$ has exactly two weak teams.
2. The probability that group $A$ has at least two weak teams.
|
\frac{1}{2}
| 39.0625 |
20,586 |
What is 0.3 less than 83.45 more than 29.7?
|
112.85
| 100 |
20,587 |
Farmer Pythagoras has now expanded his field, which remains a right triangle. The lengths of the legs of this field are $5$ units and $12$ units, respectively. He leaves an unplanted rectangular area $R$ in the corner where the two legs meet at a right angle. This rectangle has dimensions such that its shorter side runs along the leg of length $5$ units. The shortest distance from the rectangle $R$ to the hypotenuse is $3$ units. Find the fraction of the field that is planted.
A) $\frac{151}{200}$
B) $\frac{148}{200}$
C) $\frac{155}{200}$
D) $\frac{160}{200}$
|
\frac{151}{200}
| 26.5625 |
20,588 |
Given vectors $\overrightarrow{m}=(1,\sqrt{3})$, $\overrightarrow{n}=(\sin x,\cos x)$, let function $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}$
(I) Find the smallest positive period and maximum value of function $f(x)$;
(II) In acute triangle $\Delta ABC$, let the sides opposite angles $A$, $B$, $C$ be $a$, $b$, $c$ respectively. If $c=\sqrt{6}$, $\cos B=\frac{1}{3}$, and $f(C)=\sqrt{3}$, find $b$.
|
\frac{8}{3}
| 79.6875 |
20,589 |
In an acute triangle $ABC$ , the points $H$ , $G$ , and $M$ are located on $BC$ in such a way that $AH$ , $AG$ , and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$ , $AB=10$ , and $AC=14$ . Find the area of triangle $ABC$ .
|
12\sqrt{34}
| 2.34375 |
20,590 |
For how many positive integers $n$ , $1\leq n\leq 2008$ , can the set \[\{1,2,3,\ldots,4n\}\] be divided into $n$ disjoint $4$ -element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset?
|
1004
| 59.375 |
20,591 |
Given circle $O: x^2+y^2=r^2(r>0)$, $A(x_1, y_1)$, $B(x_2, y_2)$ are two points on circle $O$, satisfying $x_1+y_1=x_2+y_2=3$, $x_1x_2+y_1y_2=-\frac{1}{2}r^2$, calculate the value of $r$.
|
3\sqrt{2}
| 36.71875 |
20,592 |
Given an ellipse E: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with a focal length of $2\sqrt{3}$, and the ellipse passes through the point $(\sqrt{3}, \frac{1}{2})$.
(Ⅰ) Find the equation of ellipse E;
(Ⅱ) Through point P$(-2, 0)$, draw two lines with slopes $k_1$ and $k_2$ respectively. These two lines intersect ellipse E at points M and N. When line MN is perpendicular to the y-axis, find the value of $k_1 \cdot k_2$.
|
\frac{1}{4}
| 16.40625 |
20,593 |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?
|
2\sqrt{10}
| 25 |
20,594 |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2017,0),(2017,2018),$ and $(0,2018)$. What is the probability that $x > 9y$? Express your answer as a common fraction.
|
\frac{2017}{36324}
| 39.0625 |
20,595 |
Given $α \in \left( \frac{π}{2}, π \right)$, and $\sin α = \frac{1}{3}$.
$(1)$ Find the value of $\sin 2α$;
$(2)$ If $\sin (α+β) = -\frac{3}{5}$, and $β \in (0, \frac{π}{2})$, find the value of $\sin β$.
|
\frac{6\sqrt{2}+4}{15}
| 3.125 |
20,596 |
Let \( g(x) = 3x^4 + 2x^3 - x^2 - 4x + s \). Find the value of \( s \) such that \( g(-1) = 0 \).
|
-4
| 92.1875 |
20,597 |
Calculate the coefficient of the term containing $x^4$ in the expansion of $(x-1)(x-2)(x-3)(x-4)(x-5)$.
|
-15
| 93.75 |
20,598 |
The volume of a sphere is increased to $72\pi$ cubic inches. What is the new surface area of the sphere? Express your answer in terms of $\pi$.
|
36\pi \cdot 2^{2/3}
| 6.25 |
20,599 |
Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and the altitude on side $BC$ is $\frac{a}{2}$. Determine the maximum value of $\frac{c}{b}$.
|
\sqrt{2} + 1
| 17.1875 |
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